Nonlinear perturbations of higher dimensional anti-de Sitter spacetime

Abstract

We study nonlinear gravitational perturbations of vacuum Einstein equations, with $\Lambda <0$ in $(n+2)$ dimensions, with $n>2$, generalizing previous studies for $n=2$. We follow the formalism by Ishibashi, Kodama and Seto to decompose the metric perturbations into tensor, vector and scalar sectors, and simplify the Einstein equations. The tensor perturbations are the new feature of higher dimensions. We render the metric perturbations asymptotically anti-de Sitter by employing a suitable gauge choice for each of the sectors. Finally, we analyze the resonant structure of the perturbed equations at second order for the five dimensional case, by a partial study of single mode tensor-type perturbations at the linear level. For the cases we studied, resonant terms vanish at second order.

43 pages, discussion expanded, numerical factor missed in a Mathematica file related to previous version corrected, thanks to referees for their comments that led to correction. Resonant terms now vanish at second order for cases studied

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• Resonant Hamiltonian systems and weakly nonlinear dynamics in AdS spacetimes